where the scale factor \(a(t)=a_0t^{\ell } \) , \(\ell \in {\mathbb {R}}\) , \( t>0\) , and \(x \in {\mathbb {R}}^3\) , \(x_0=t \) . If \(\ell <0\)  the spacetime is contracting. In the case of \(\ell >1\)  the expansion is accelerating (with horizon), while for \(0<\ell <1\)  the expansion is decelerating. In the case of the Milne spacetime [1]}, [2]}, [3]} \(\ell =1 \) .
The FLRW spacetime with the scale factors \(a(t)=a_0t^{2/3} \)  and \(a(t)=a_0t^{1/2} \)  (see, e.g., [4]}, [5]}) are modeling the matter dominated universe (Einstein & de Sitter model [6]}) and the radiation dominated universe, respectively. The Dirac equation in the spacetime with the metric tensor (REF ) is (see, e.g., [7]})
\(\displaystyle \left( i {\gamma }^0 \partial _t +i \frac{1}{a(t)}{\gamma }^1 \partial _{x_1}+i \frac{1}{a(t)}{\gamma }^2 \partial _{x_2}+i \frac{1}{a(t)}{\gamma }^ 3 \partial _{x_3} +i \frac{3\dot{a}(t)}{2a(t)} {\gamma }^0 -m{\mathbb {I}}_4 \right)\Psi =F \,,\) 
